Gary Koop This Page Intentionally Left Blank Bayesian Econometrics Gary Koop To Lise Contents Preface xiii 1 An Overview of Bayesian Econometrics 1 For instance, Arnold Zellner’s seminal Bayesian econometrics book ( Zellner. Bayesian Econometrics introduces the reader to the use of Bayesian methods in the field of Gary Koop is Professor of Economics at the University of Glasgow. A working paper which describes a package of computer code for Bayesian VARs The BEAR Toolbox by Alistair Dieppe, Romain Legrand and Bjorn van Roye.

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The interested reader is referred to Poirierwhich provides a deeper discussion of this issue and includes an extensive bibliography. Regardless of how a researcher feels about prior information, it should in no way be an obstacle to the adoption of Bayesian methods.

For instance, in the linear regression model which will be discussed in the next chapterit is common to assume that the errors have a Normal distribution. However, in economics we typically work with models bahesian depend upon parameters.

The first of these is an independent Normal-Gamma prior and the second a natural conjugate prior subject to inequality restrictions. As discussed in that section, Monte Carlo integration is not required for the Normal linear regression model with natural conjugate prior, unless interest centers on nonlinear functions of the regression coefficients.

Since there are five unknown regression coefficients, the answers to five questions of this form would give the researcher five equations in econometrucs unknowns. These use data-based information to choose the prior and, hence, violate a basic premise of Bayesian methods. In many cases, it is reasonable to assume that returns to scale are roughly constant. A researcher who wishes to be noninforma- tive about 9 would allocate equal prior weight to each equally sized sub-interval e.

As we will discuss in subsequent chapters, there are special techniques in many cases for calculating the posterior odds ratio directly. However, in cases where different researchers can approach a problem with very different priors, a Bayesian analysis with only a single prior can be criticized.

As described in Chapter 1, even if we do not economefrics the properties e. If we use this method to work out the predictive density of the sales price of a house with a lot size of square feet, two bedrooms, two bathrooms and one storey, we find the predictive mean and standard deviation to be and 18respectively. A straightforward implication of the law of large numbers given in Appendix B see Definition B.

The practical impor- tance of these two differences is that, typically, more draws are required with Gibbs sampling than with Monte Carlo integration to achieve econojetrics given level of accuracy.

Hence, we do not present posterior odds ecohometrics using the noninforma- tive prior. This may imply a redefinition of the dependent variable. It sum- marizes all we know about 9 after i.

As described in Section 3. These algorithms will be used in later chapters. The problem with such a motivation is that it is typically not possible to find such an initial draw.

### SGPE: Bayesian Econometrics – Gary Koop

That is, Mj for j — 1, 2 is based on the simple linear regression model: The purpose of this question is to learn about the properties of the Gibbs sampler in a very simple case. Economettics is common to present highest posterior density intervals in addition to point estimates when doing Bayesian estimation. Clearly, this is unreasonable and provides a strong argument for saying that informative priors should always be used for and fpat least for coefficients that are not common to both models.

The first of these computational methods is Gibbs sampling. In most cases, it is not possible to work with all these building blocks analytically.

Hence, if you can successfully find 0 g, then sequentially drawing from the posterior of 9 p conditional on the previous draw for 9 2then 9 2 given the previous draw for 9 p, will yield a sequence of draws from the posterior. There are many ways of gauging the approximation error associated with a particular value of S. Accordingly, in addition to discussing computation in detail in the book itself, the website associated with the book contains MATLAB programs for performing Bayesian analysis in a wide variety of models.

For instance, MATLAB code relating to the following empirical illustration is available on the website associ- ated with this book. For instance, interest may center on whether a particular parameter is positive. By dropping out this middle set, we make it likely that our first and last set of draws are independent of one another.

In this case, the econometrician might be interested in finding out whether the advertising campaign increased sales i. By set- ting g. Linear Regression Model with a Single Variable 23 In one sense, this noninformative prior has very attractive properties and, given the close relationship with OLS results, provides a bridge between the Bayesian and frequentist approaches. Monte Carlo Integration 46 3.

The second computational method introduced is importance sampling.

## Bayesian Econometrics

It is more technically correct to call draws generated by the computer pseudo-random. For the reader who knows time series methods, S’fO is the spectral density of the sequence 9 1 ‘- 1 for. Chib and Greenberg is an excellent introduction to the Metropo- lis-Hastings algorithm and includes a derivation of the acceptance probability which ensures that the Metropolis-Hastings algorithm converges to bayeslan poste- rior.

For the regression coefficients, we set: